Integer is Congruent Modulo Divisor to Remainder
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Theorem
Let $a \in \Z$.
Let $a$ have a remainder $r$ on division by $m$.
Then:
- $a \equiv r \pmod m$
where the notation denotes that $a$ and $r$ are congruent modulo $m$.
Corollary
$a \equiv b \pmod m$ if and only if $a$ and $b$ have the same remainder when divided by $m$.
Proof
Let $a$ have a remainder $r$ on division by $m$.
Then:
- $\exists q \in \Z: a = q m + r$
Hence by definition of congruence modulo $m$:
- $a \equiv r \pmod m$
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 14.2 \ \text{(i)}$: Congruence modulo $m$ ($m \in \N$)